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City University of New York (CUNY) City University of New York (CUNY)

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Dissertations, Theses, and Capstone Projects CUNY Graduate Center

9-2016

On Sums of Binary Hermitian Forms On Sums of Binary Hermitian Forms

Cihan Karabulut

The Graduate Center, City University of New York

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On Sums of Binary Hermitian Forms

by

Cihan Karabulut

A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfill- ment of the requirements for the degree of Doctor of Philosophy, The City University of New York.

2016

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ii

2016 c

Cihan Karabulut

All Rights Reserved

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iii This manuscript has been read and accepted for the Graduate Faculty in Mathe- matics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy.

Gautam Chinta

Date Chair of Examining Committee

Ara Basmaijan

Date Executive Officer

Gautam Chinta

Carlos Moreno

Cormac O’Sullivan

Supervisory Committee

THE CITY UNIVERSITY OF NEW YORK

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iv Abstract

On Sums of Binary Hermitian Forms by

Cihan Karabulut

Advisor: Gautam Chinta

In one of his papers, Zagier defined a family of functions as sums of powers of

quadratic polynomials. He showed that these functions have many surprising prop-

erties and are related to modular forms of integral weight and half integral weight,

certain values of Dedekind zeta functions, Diophantine approximation, continued

fractions, and Dedekind sums. He used the theory of periods of modular forms to

explain the behavior of these functions. We study a similar family of functions,

defining them using binary Hermitian forms. We show that this family of functions

also have similar properties.

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Acknowledgements

I would like to express my sincere gratitude to my advisor, Gautam Chinta, for his constant support and guidance throughout the completion of this thesis. He has been incredibly patient and given me timely advice and encouragement whenever I needed. I am especially thankful to him for first suggesting the problem and for providing the Sage code, which provided evidence that results similar to Zagier’s may also be true in our case.

I would like to thank Carlos Moreno and Cormac OSullivan for serving on my defense committee and for their comments and suggestions that improved this thesis.

I am also grateful for the support of the faculty and staff of the Graduate Center:

Jozef Dodziuk, Rob Landsman, and many others.

I would like to thank those colleagues and fellow graduate students with whom I discussed mathematics and formed sustaining friendships. In particular, I would like to thank Jorge Basilio, Jorge Florez, Tian An Wong, Bora Ferlengez, Ozgur Evren, Carlos Arreche, Sajjad Lakzian, Cheyne Miller, Byungdo Park, Joe Kramer Miller, Zachary McGuirk, and Tianyi Mao.

v

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vi

Finally, I would like to thank my family. My father, who himself never had any

formal education, always supported, encouraged, and instilled in me a love of nature,

philosophy, and mathematics. I am grateful to my parents-in-law for providing excel-

lent childcare, home-cooked meals, and a home away from my native home. Most of

all, I would like to thank my wonderful wife for whom this small space is insufficient

to contain adequate gratitude. Without her love and support none of this would

have been possible.

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Contents

1 Introduction 1

1.0.1 History and motivation . . . . 1

1.0.2 Main results . . . . 3

1.0.3 Future directions . . . . 5

2 Zagier’s Function 8 2.1 Connection to continued fractions . . . . 9

2.2 The modular connection . . . . 11

2.2.1 Eichler-Shimura Isomorphism . . . . 11

2.2.2 Periods of modular forms . . . . 13

3 Sums of binary Hermitian forms 19 3.1 Binary Hermitian forms . . . . 19

3.2 Sums of binary Hermitian forms over PSL(2, Z[i]) . . . . 22

vii

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CONTENTS viii

4 Continuity of H

k,d

and Hurwitz’s algorithm 39

4.1 Hurwitz’s continued fraction algorithm . . . . 41 4.2 Continuity of H

2,d

. . . . 45

Bibliography 49

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Chapter 1 Introduction

1.0.1 History and motivation

In [18], Zagier defines a family of functions that are defined as sums of powers of certain quadratic polynomials with integer coefficients and fixed discriminant, and he discovers that these functions have many surprising properties. Specifically, Zagier considers the following family of functions:

For non-square discriminant D ≡ 0, 1 (mod 4) and k a positive even integer, define a function F

k,D

: R → R as follows,

F

k,D

(x) := X

disc(Q)=D a<0<Q(x)

Q(x)

k−1

(1.1)

where Q(X) = aX

2

+ bX + c with (a, b, c) ∈ Z

3

and disc(Q) = b

2

− 4ac. When D is a square discriminant, one has to add a simple correction term consisting of the k-th Bernoulli polynomial. But for the sake of simplicity, let us assume that D is a non square discriminant.

1

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CHAPTER 1. INTRODUCTION 2

For instance, Zagier finds that F

2,5

is constant with value F

2,5

(x) = 2 for any x ∈ R despite the fact that there are infinitely many Q’s contributing to the sum for F

2,5

when x is irrational. More generally, F

2,D

(x) = −5L(−1, χ

D

) where L(−1, χ

D

) is the Dirichlet L-series of the character χ

D

(n) =  D

n



(Kronecker symbol). For k = 4, Zagier shows that F

4,D

(x) = L(−3, χ

D

), but for k ≥ 6 F

k,D

is no longer constant because of the existence of cusp forms of weight 2k on the modular group SL(2, Z). More precisely, for k ≥ 6, the function F

k,D

is a linear combination of a constant function and the functions X

n≥1

a

f

(n)

n

2k−1

cos(2πnx), where f runs over the normalized Hecke eigenforms in the space of cusp forms of weight 2k for the modular group and a

f

(n) denotes the n-th Fourier coefficient of f .

The function F

k,D

(x) is related to a family of cusp forms f

k,D

, defined in chapter

2, that Zagier introduced in [17]. These cusp forms were later studied by Kohnen

and Zagier ([10], [12]) in connection with Doi-Naganuma lifting from modular forms

to Hilbert modular forms and in connection with Shimura-Shintani correspondence

between modular forms of integral and half-integral weight. Kohnen and Zagier

[11] also used f

k,D

as an example of modular forms whose periods are rational and

arithmetically very interesting. The precise relation between the function F

k,D

and

f

k,D

is that F

k,D

is the even part of the Eichler integral, also defined in chapter 2, of

f

k,D

on R and so gives the even part of the period polynomial of f

k,D

. This relation

beautifully explains a natural question that might have occurred to the reader by

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CHAPTER 1. INTRODUCTION 3

now as to why F

k,D

is constant for k = 2, 4 but not for k ≥ 6.

The function F

k,D

is also related to continued fraction expansion of real numbers, modular forms of half integral, and Dedekind sums.

1.0.2 Main results

In this thesis, we define a family of functions, analogous to F

k,D

, from C to R by replacing the quadratic polynomials with functions coming from binary Hermitian forms over the Gaussian integers. That is, for d a positive integer that is not a norm in Z[i] with d ≡ 0, 1, 2 (mod 4) and k a positive even integer, define a function H

k,d

: C → R as

H

k,d

(z) := X

∆(h)=d a<0<h(z)

h(z)

k−1

(1.2)

where h(z) = az ¯ z + 1

2 (bz + ¯ b¯ z) + c, ∆(h) = |b|

2

− 4ac, and b ∈ Z[i]. The function h(z) in the sum comes from evaluating the binary Hermitian form h(z, w) := az ¯ z + 1

2 (bz ¯ w + ¯ b¯ zw) + cw ¯ w attached to the binary Hermitian matrix h =

 a b/2

¯ b/2 c



over

Z[i] at w = 1. The reason we choose the functions h(z) to replace Q’s in equation 1.1

is because quadratic polynomials with integer coefficients are just binary quadratic

forms with one of the argument equal to one, and essentially one can think of binary

Hermitian forms as binary quadratic forms over the rings of imaginary quadratic

fields. More importantly, similar to the action of SL(2, Z) on binary quadratic forms,

we also have an action of SL(2, O), which allows us to prove results similar to Zagier’s.

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CHAPTER 1. INTRODUCTION 4

To state our main results, let us define the following numbers, which are the values of H

k,d

at z = 0:

ω

k,d

: = X

b21+b22−4ac=d a<0<c

c

k

(1.3)

= X

b21+b22<d b1+b2≡d(mod 2)

σ

k

 d − b

21

− b

22

4



(1.4)

where b = b

1

+ ib

2

∈ Z[i].

Theorem 1. For every positive integer d ≡ 0, 1, 2 (mod 4) that is not a norm of a Gaussian integer, H

2,d

(z) and H

4,d

(z) have constant values ω

2,d

and ω

4,d

respectively.

We prove these results using the continuity of H

k,d

and the functional equations satisfied by H

k,d

. It is straight forward to prove the functional equations and the continuity when k ≥ 4 but when k = 2 the proof of the continuity is non-trivial.

To prove the continuity in the case k = 2, we use a continued fraction algorithm for complex numbers using Gaussian integers discovered by Hurwitz in [8]. So, our function like F

k,D

is related to the continued fraction expansion of complex numbers.

Theorem 1 shows that the vector space spanned by H

2,d

and H

4,d

as d varies is

one dimensional. Using the same ideas as in the proof of 1, we are able to show

for k = 6 that the space spanned by H

6,d

is two dimensional and in general finite

dimensional for fixed k and varying d. That is,

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CHAPTER 1. INTRODUCTION 5

Theorem 2. H

6,d

(z) = κ

d

H

6,6

(z) + (ω

6,d

− κ

d

ω

6,6

), where κ

d

is a rational number depending on d.

Theorem 3. H

k,d

(z) is a linear combination of finitely many H

k,dj

where d

j

is a positive integer that is not a norm in Z[i] with d

j

≡ 0, 1, 2 (mod 4) for each j.

1.0.3 Future directions

The main question that arises from this thesis is: Is there a modular explanation for the behavior of H

k,d

? If so, what kind of a modular object is it? We suspect the function H

k,d

is also related to an automorphic object, and we suspect this automorphic object to be a Bianchi modular form. One can view the Bianchi modular forms as modular forms over imaginary quadratic fields. However, when the field is an imaginary quadratic field some difficulties arise, since the symmetric space one has to consider is the hyperbolic 3-space H

3

which has no complex structure. Despite this fact, many of the fundamental arithmetical results known for the classical modular forms are believed to be true for Bianchi modular forms. For example, there is computational evidence of the Shimura-Taniyama-Weil type correspondence between elliptic curves and Bianchi modular forms defined over imaginary quadratic fields.

For such an example, we refer the reader to [3] and [16].

Specifically, Bianchi modular forms are certain real analytic functions on H

3

that

transform like modular forms with respect to the action of PSL(2, O

K

) on H

3

where

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CHAPTER 1. INTRODUCTION 6

O

K

is the ring of integers of the imaginary quadratic field K.

They can also be defined as certain classes in the cohomology of quotients of H

3

by congruence subgroups of PSL(2, O

K

). Similar to classical modular forms, there is a version of the Eichler-Shimura isomorphism, called Eichler-Shimura-Harder iso- morphism, for Bianchi modular forms. Eichler-Shimura-Harder isomorphism states that

H

cusp1

(Γ, V

k,k

(C)) ∼ = S

k

(Γ) (1.5)

where Γ is a congruence subgroup in PSL(2, O

K

) and V

k,k

(C)) := V

k

(C)⊗

C

V

k

(C) with V

k

(C) denoting the space of homogeneous polynomials of degree k in two variables and V

k

(C) is its twist by complex conjugation.

We review the classical Eichler-Shimura isomorphism in chapter 2 and show how

this isomorphism relates F

k,D

to the cusp form f

k,D

. We have done some work on

the cocycle property of H

k,d

for PSL(2, Z[i]) and shown that the cocycle one gets

from H

k,d

is related to the parabolic 1-cocycles, which live in the cuspidal cohomol-

ogy of PSL(2, Z[i]). Consequently, we are led to believe that there is a connection

between H

k,d

and Bianchi modular forms. This work is still in progress, and at the

moment we are trying to compute the dimension of the space of parabolic 1-cocycles

of PSL(2, Z[i]). Finding the dimension of this space will also give us the dimension of

the space spanned by the functions H

k,d

for k fixed and varying d. But ultimately, to

establish a relationship between H

k,d

and Bianchi modular forms, we need to adjust

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CHAPTER 1. INTRODUCTION 7

or find analogs of the other two key ingredients, Bol’s identity and Eichler integral, which we define in chapter 2, used to relate F

k,D

and f

k,D

.

A second task is to consider H

k,d

over the rings of integers of imaginary quadratic fields different than Gaussian integers. The definition of H

k,d

makes sense over other rings, since binary Hermitian forms are defined over the rings of imaginary quadratic fields.

The organization of this thesis is as follows. In chapter 2, we recall some of Za-

gier’s results and review the F

k,D

’s connection to continued fractions and to modular

forms in a fair amount of detail. In chapter 3, after stating the relevant facts about

binary Hermitian forms, we define our function H

k,d

and prove the Theorems 1, 2

and 3. In chapter 4, we discuss the continuity of H

k,d

. We first prove the continuity

for k > 4. To prove the continuity for k = 2, we use the Hurwitz continued fraction

algorithm. We first describe the algorithm as it is usually stated, but for our pur-

poses we recast the algorithm in terms of matrix notation. This allows us to describe

the functions appearing in the sum for H

2,d

for a given z ∈ C in terms of the action

of the matrices built from the continued fraction expansion of z. These functions

decay to zero exponentially quickly, which proves the continuity of H

2,d

.

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Chapter 2

Zagier’s Function

For non-square D ≡ 0, 1 (mod 4) and k a positive even integer, Zagier [18] defines the following function:

F

k,D

(x) := X

disc(Q)=D a<0<Q(x)

Q(x)

k−1

(2.1)

where Q(X) = aX

2

+ bX + c with (a, b, c) ∈ Z

3

and disc(Q) = b

2

− 4ac.

Zagier shows that,

Theorem 4. Let D be a positive fundamental discriminant. Then the sums defining F

2,D

and F

4,D

converge for all x ∈ R and have constant values α

D

and β

D

respectively independent of x where

α

D

= −5L(−1, χ

D

) and β

D

= L(−3, χ

D

).

For k ≥ 6, F

k,D

is no longer constant but instead is a linear combination of a finite collection of functions. More precisely, Zagier proves

8

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CHAPTER 2. ZAGIER’S FUNCTION 9

Theorem 5. For every positive integer k and every positive non-square discriminant D, the function F

k,D

is a linear combination, with coefficients depending on D and k, of a finite collection of functions depending only on k.

Zagier proves these results using the functional equations

F

k,D

(x + 1) = F

k,D

(x), x

2k−2

F

k,D

(1/x) − F

k,D

(x) = P

k,D

(x) (2.2)

where

P

k,D

(x) = X

b2−4ac=D a>0>c

(ax

2

+ bx + c)

k−1

, (2.3)

and the continuity of F

k,D

. The functional equations are easy to prove when x ∈ Q.

The continuity of F

k,D

is also easy to prove when k ≥ 4. But for k = 2, Zagier only shows that F

2,D

is convergent and states that the continuity can be proved in an elementary way, but he doesn’t give the proof. He conjectures that the quadratic functions appearing in the sum when k = 2 for a given value of x are related to the continued fraction expansion of x.

2.1 Connection to continued fractions

The convergence of F

k,D

is not immediate at all. Zagier observes that one has Q(x) = O(1/a

Q

) for all Q in (2.1) and the number of contributing Q’s per value of a is bounded. Therefore the series (2.1) converges at most like

X

a=1

1

a

k−1

. So, the

sum defining F

k,D

converges uniformly when k ≥ 4 but this argument fails when

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CHAPTER 2. ZAGIER’S FUNCTION 10

k = 2. However, experiments carried out by Zagier for the value x = 1

π suggest that the sum defining F

2,5

converges extremely rapidly. Furthermore, Zagier observes a strong pattern among the Q’s appearing in the sum for the value x = 1

π and D = 5 which explains the rapid convergence of F

2,5

. Based on this observation he makes the following conjecture. To state the conjecture we need some notation.

Let x be a real number between 0 and 1 and write x as a continued fraction

x = 1

n

1

+ 1 n

2

+ 1

. ..

= [0, n

1

, n

2

, . . .] (2.4)

using integers n

1

, n

2

, . . . ≥ 1 which are called partial quotients. The continued frac- tion expansion terminates for x if and only if x ∈ Q, so assume x is irrational. Define the real numbers δ

0

, δ

1

, δ

2

, . . . inductively by

δ

0

= 1, δ

1

= x, δ

j+1

= δ

j−1

− n

j

δ

j

(j ≥ 1). (2.5)

Zagier observes the following based on numerical evidence:

Conjecture 1. The functions appearing in the sum for F

2,5

 1 π



are all of the expressions δ

2j

+ δ

j

δ

j+1

− δ

2j+1

and some of the expressions δ

j2

− δ

j

δ

j+1

− δ

2j+1

where the linear forms δ

j

∈ Z + Zx are defined as above by the continued fraction expansion of x = 1

π .

The conjecture shows why F

2,5

converges rapidly; δ

j

decays exponentially in j.

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CHAPTER 2. ZAGIER’S FUNCTION 11

In fact, we have the following uniform bound on δ

j

’s independent of x; δ

j

< 2

1−j2

. Therefore the sum converges uniformly and is a continuous function.

This conjecture was recently proved by P. Bengoechea, who was a PhD student of Zagier, in [1] for any fundamental discriminant D. Her results also provide a proof of continuity when k = 2. The same conjecture was also proved by M. Jameson, who was a PhD student of K. Ono, in [9] for any D. But, she assumes the continuity of F

2,D

.

2.2 The modular connection

A natural question to ask is why F

k,D

is constant when k = 2, 4 but not constant for k ≥ 6. The short answer is that there are cusp forms of weight 2k on the full modular group for k ≥ 6 but not for k < 6. We will expand on this answer by exploring the relation between F

k,D

and modular forms.

2.2.1 Eichler-Shimura Isomorphism

Let Γ = PSL(2, Z) and V

k

be the space of polynomials in X of degree less than or equal to k − 2 over C. Let |

2−k

denote the usual weight 2 − k action of Γ on V

k

defined by

(P |

2−k

γ)(X) = (cX + d)

k−2

P  aX + b cX + d

 

P (X) ∈ V, γ = a b c d



. (2.6)

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CHAPTER 2. ZAGIER’S FUNCTION 12

Under the action |

2−k

, V

k

splits into the direct sum of subspaces V

k+

and V

k

of even and odd polynomials respectively. The action |

2−k

on V

k

can be extended by linearity to the action of the group ring Z[Γ].

A 1-cocycle with values in V

k

is a map ϕ : Γ → V

k

such that

ϕ(γ

1

γ

2

) = ϕ(γ

1

)|

2−k

γ

2

+ ϕ(γ

2

) (2.7)

where γ

1

, γ

2

∈ Γ. We let Z(Γ, V

k

) denote the group of all 1-cocycles with values in V

k

. Given fixed v ∈ V

k

, it is clear that the map

γ 7→ v|

2−k

(1 − γ) (2.8)

for all γ ∈ Γ defines a cocycle. A cocycle of this form is called a coboundary. We let B(Γ, V

k

) to denote the group of all coboundaries with values in V

k

. A 1-cocyle ϕ is called cuspidal or parabolic if

ϕ(T ) = v|

2−k

(1 − T ) (2.9)

for some v ∈ V

k

, where T = 1 1 0 1



. The group of cuspidal 1-cocyles with values in V

k

is denoted by Z

1

(Γ, V

k

). Note that the group of all cuspidal 1-cocycles contains the group of coboundaries since if ϕ ∈ B(Γ, V

k

) then by definition ϕ(T ) = v|

2−k

(1 − T ).

The factor group

H

1

(Γ, V

k

) = Z

1

(Γ, V

k

)

B(Γ, V

k

)

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CHAPTER 2. ZAGIER’S FUNCTION 13

is called the Eichler cohomology group. This group sometimes is also called the cuspidal cohomology and denoted by H

cusp1

(Γ, V

k

)

Eichler-Shimura isomorphism states that (cf. [13], [15], [5], [11] )

S

k

⊕ S

k

∼ = H

1

(Γ, V

k

) (2.10)

where S

k

is the space of cusp forms of weight k on Γ and S

k

is the space of antiholo- morphic cusp forms consisting of functions ¯ f (z) := f (z) with f ∈ S

k

. The map used to show the isomorphism in (2.10) is called the period map. The Eichler-Shimura isomorphism implies that there exists cusp forms all of whose even and odd periods are rational. Kohnen and Zagier [11] give examples of such cusp forms whose pe- riods are arithmetically interesting expressions – relating to Bernoulli numbers, to binary quadratic forms, to zeta functions of real quadratic fields, to modular forms of half-integral weight, and to Hilbert modular forms. F

k,D

is related to the even period polynomial of one of the cusp forms they study in [11].

2.2.2 Periods of modular forms

In this section we describe two maps; one from S

k

to V

k

and another from S

k

to

H

1

(Γ, V

k

). The Eichler-Shimura isomorphism can be stated using either one of these

maps.

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CHAPTER 2. ZAGIER’S FUNCTION 14 Let f be a cusp form of weight k. One defines the nth period of f by

r

n

(f ) = Z

0

f (it)t

n

dt (2.11)

where 0 ≤ n ≤ k − 2. The period polynomial of f is defined to be

r(f )(X) = Z

i∞

0

f (z)(X − z)

w

dz =

w

X

n=0

i

−n+1

w n



r

n

(f )X

w−n

(2.12)

where 0 ≤ w ≤ k − 2. It is clear that r(f )(X) ∈ V

k

. We also set

r

+

(f ) = X

0≤n≤w n even

(−1)

n/2

w n



r

n

(f )X

w−n

, (2.13)

r

(f ) = X

0<n<w n odd

(−1)

(n−1)/2

w n



r

n

(f )X

w−n

, (2.14)

so that r

±

∈ V

k±

, r = ir

+

+ r

. It is shown in [11] that

r(f )|

2−k

(1 + S) = 0, (2.15)

r(f )|

2−k

(1 + U + U

2

) = 0 (2.16)

so that r(f ) belongs to the subspace W

k

of V

k

defined by

W

k

= ker(1 + S) ∩ ker(1 + U + U

2

) (2.17)

= {P ∈ V

k

: P |

2−k

(1 + S) = P |

2−k

(1 + U + U

2

) = 0} (2.18)

where S = 0 −1 1 0



and U = 1 −1 1 0



. Under the action, W

k

also splits as W

k

=

W

k+

⊕ W

k

. We thus have two maps r

±

: S

k

→ W

k±

. In this setting the statement of

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CHAPTER 2. ZAGIER’S FUNCTION 15 Eichler-Shimura isomorphism becomes:

Theorem 6. The map r

: S

k

→ W

k

is an isomorphism. The map r

+

: S

k

→ W

k+

/ < X

k−2

− 1 > is an isomorphism.

Alternatively, one can define the period mapping f 7→ r(f ) directly as a map from S

k

to H

1

(Γ, V

k

) by using the so-called “Eichler integral” of f . If f (z) =

X

n=1

a

n

(f )q

n

(q = e

2πiz

) is in S

k

then the (k − 1)-fold integral of f which we denote by ˜ f is given by

f (z) = ˜ 1 (2πi)

k−1

X

n=1

a

n

(f )

n

k−1

q

n

. (2.19)

We call ˜ f the Eichler integral of f . Note that ˜ f is unique up to adding some element of V

k

.

Let F be a holomorphic function defined on the upper-half plane H. The Bol’s identity [2] states that,

d

k−1

dz

k−1

(F |

2−k

γ) =  d

k−1

F dz

k−1



|

k

γ (2.20)

where (F |

k

)(z) = (cz + d)

−k

F (γz) for γ = a b c d

 . If we let F = ˜ f then we have

d

k−1

dz

k−1

( ˜ f |

2−k

(1 − γ)) = d

k−1

f ˜ dz

k−1

!

|

k

(1 − γ) = 1

(2πi)

k−1

(f − f |

γ

) = 0 (2.21) Thus, the function

r

f,γ

(z) :=  ˜ f |

2−k

(1 − γ) 

(z)

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CHAPTER 2. ZAGIER’S FUNCTION 16

is a polynomial in z of degree ≤ k−2. The map γ 7→ r

f,γ

from Γ to V

k

is automatically a cocycle since it is a coboundary in the space of functions on H. Therefore we get r

f,γ

∈ H

1

(Γ, V

k

) and this element doesn’t depend on the choice of ˜ f . To see this, suppose we chose another Eichler integral ˜ g of f then the difference of ˜ f − ˜ g is a polynomial P ∈ V

k

and so

r

f,γ

− r

g,γ

=  ˜ f |

2−k

(1 − γ) 

− (˜ g|

2−k

(1 − γ)) = P |

2−k

(1 − γ)

is a coboundary. In [5], M. Eichler proves the isomorphism in 2.10 using the Eichler integral.

The Eichler integral of f ∈ S

k

also converges on the real line and there it can be split into even and odd parts as follows

f ˜

+

= 1 (2πi)

k−1

X

n=1

a

n

(f )

n

k−1

cos(2πnx), f ˜

= 1 (2πi)

k−1

X

n=1

a

n

(f )

n

k−1

sin(2πnx).

Using ˜ f

+

, ˜ f

one can split r

f,γ

into its even and odd parts as

r

+f,γ

(x) :=  ˜ f

+

|

2−k

(1 − γ) 

(x), r

f,γ

(x) :=  ˜ f

|

2−k

(1 − γ)  (x)

where r

f,γ+

∈ W

k+

/ < X

k−2

− 1 > and r

f,γ

∈ W

k

.

Now, we can finally give the precise explanation for the behavior of F

k,D

. The

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CHAPTER 2. ZAGIER’S FUNCTION 17 function F

k,D

arises from studying the function

f

k,D

(z) := C

k

D

k−1/2

X

b2−4ac=D a,b,c∈Z

1

(az

2

+ bz + c)

k

(z ∈ H)

where C

k

is a normalizing factor, D ≡ 0, 1 (mod 4) and k is a positive even integer.

This function is introduced in [17] and shown to be a cusp form of weight 2k for the full modular group. It is studied in connection with Doi-Naganuma lifting from elliptic to Hilbert modular forms, and shown in ([10], [12]) to be the Dth Fourier co- efficient of the holomorphic kernel function for the Shimura-Shintani correspondence between modular forms of integral and half-integral weight. It is also conjectured that for fixed k, as D varies, the f

k,D

’s span the space of cusp forms of weight 2k.

The function f

k,D

is an example of a cusp form whose periods are arithmetically interesting expressions. The even period polynomials of f

k,D

are computed in [11]

and are given by

r

+

(f

k,D

)(x) = ζ

D

(1 − k)

2ζ(1 − 2k) (x

2k−2

− 1) − x

2k−2

F

k,D

 1 x



− F

k,D

(x).

Since r

+

(f

k,D

) ∈ W

k+

/ < X

k−2

−1 >, this implies that F

k,D

has the Fourier expansion

F

k,D

(x) = ζ

D

(1 − k) 2ζ(1 − 2k) +

X

n=0

a

k,D

(n)

n

2k−1

cos(2πnx)

where the a

k,D

(n) are the Fourier coefficients of f

k,D

which is of weight 2k. But since

there are no cusp forms below the weight 12, the sum vanishes for k = 2, 4 and the

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CHAPTER 2. ZAGIER’S FUNCTION 18

constancy of F

k,D

follows.

(28)

Chapter 3

Sums of binary Hermitian forms

Throughout this chapter, we use the following terminology and notation

• k > 0 an even integer

• A

t

is transpose of A

3.1 Binary Hermitian forms

Let A be a 2 × 2 matrix with entries in C. A is said to be a Hermitian matrix if

A = ¯ A

t

(3.1)

where ¯ A

t

is obtained from A by applying the complex conjugation to each of the entries and then taking the transpose or vice versa. Let R be a subring of C which is closed under conjugation, i.e. R = ¯ R. We denote by H(R) the set of all 2 × 2 Hermitian matrices with entries in R. Trivially, h ∈ H(R) if and only if h = a b

¯ b c



with a, c ∈ R ∩ R and b ∈ R. Every h ∈ H(R) defines a binary Hermitian form with

19

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 20

coefficients in R. If h = a b

¯ b c



the associated binary Hermitian form is the map h : C × C → R defined by,

h(z, w) = z w  a b

¯ b c

  ¯ z

¯ w



= az ¯ z + bz ¯ w + ¯ b¯ zw + cw ¯ w.

We shall often call an element h ∈ H(R) a binary Hermitian form with coefficients in R. The discriminant ∆(h) of h ∈ H(R) is defined as

∆(h) = −4det(h). (3.2)

The study of these forms parallels the study of binary quadratic forms and they are known to be related to Bianchi modular forms (cf. [6],[7]). Similar to the action of the GL(2, Z) on binary quadratic forms, we have the action of GL(2, R) on H(R) given by the formula

σ(h) = σh¯ σ

t

(3.3)

where σ ∈ GL(2, R) and h ∈ H(R). We note that ∆(σ(h)) = |det σ|

2

· ∆(h). A binary Hermitian form h ∈ H(R) is positive definite if h(u, v) > 0 for all (u, v) ∈ C × C\{(0, 0)} and h is called indefinite if ∆(h) < 0. We let

H

+

(R) = {h ∈ H(R)

h is positive definite}, H

(R) = {h ∈ H(R)

h is indefinite}.

Note that H

±

(R) is invariant under the action of the group GL(2, R).

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 21 Let K = Q( √

−D) be the imaginary quadratic number field of discriminant D and let O be its ring of integers. The study of the orbits of H(O) under the action of SL(2, O) is called the reduction theory of binary Hermitian forms.

Definition 1. Let d ∈ Z be an integer. We define

H(O, d) = {h ∈ H(O)

∆(h) = d}, H

±

(O, d) = {h ∈ H

±

(O)

∆(h) = d}.

Similar to the main result of reduction theory of binary quadratic forms we also have the following result for binary Hermitian forms.

Theorem 7. For any d ∈ Z with d 6= 0 the sets H(O, d) and H

±

(O, d) split into finitely many SL(2, O) orbits.

Proof. cf. [6] §9.2, Theorem 2.2.

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 22

3.2 Sums of binary Hermitian forms over PSL(2, Z[i])

Using binary Hermitian forms we define a function analogous to F

k,D

. Let h

 a b/2

¯ b/2 c



∈ H(Z[i]). We define the function H

k,d

: C → R as

H

k,d

(z) : = X

∆(h)=d a<0<h(z,1)

h(z, 1)

k−1

(3.4)

= X

|b|2−4ac=d a<0

max 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



k−1

!

(3.5)

where d is a positive integer that is not a norm in Z[i] with d ≡ 0, 1, 2 (mod 4). We will discuss the convergence and continuity in the next chapter. Let us assume for now that H

k,d

converges and is continuous for all k, d and for all z ∈ C.

One can think of each binary Hermitian form h(z, 1) appearing in (3.4) as a polynomial in z and ¯ z. There is a natural action of PSL(2, Z[i]) on the space of such polynomials and this action is compatible with the action in (3.3). It is essentially one variable version of the action on the space of homogeneous polynomials in z, w tensored over C with the space of homogeneous polynomials in ¯ z, ¯ w. This space is used when one generalizes the Eichler-Shimura isomorphism for modular forms over imaginary quadratic fields which are called Bianchi modular forms.

Let V

k

be the space of polynomials in z and ¯ z of degree at most k − 1 in both z

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 23 and ¯ z over C, i.e.

V

k

= (

P (z) = X

0≤n,j≤k−1

a

n,j

z

n

z ¯

j

: a

n,j

∈ C )

.

PSL(2, Z[i]) acts on V

k

by,

(P |

1−k

γ)(z) = (cz + d)

k−1

(cz + d)

k−1

P  az + b cz + d



(3.6)

where γ = a b c d



. Note that the set of all h(z, 1) appearing in (3.4) is a subset of V

2

. The next proposition gives the relation between the two actions (3.3) and (3.6) when P ∈ V

2

is a binary Hermitian form.

Proposition 1. Let P (z) = az ¯ z + bz + ¯ b¯ z + c = z 1  a b

¯ b c

  ¯ z 1



∈ V

2

. Then

(P |

−1

γ)(z) = z 1 γ

t

a b

¯ b c



¯ γ  ¯ z

1

 .

Proof. Let γ = p q r s



. Then

(P |

−1

γ)(z) = (rz + s)(rz + s)P  pz + q rz + s



= a(pz + q)(pz + q) + b(pz + q)(rz + s) + ¯ b(pz + q)(rz + s) + c(rz + s)(rz + s)

= pz + q rz + s  a b

¯ b c

 pz + q rz + s



= z 1  p r q s

 a b

¯ b c

  ¯ p q ¯

¯ r s ¯

  ¯ z 1



= z 1 γ

t

a b

¯ b c



¯ γ  ¯ z

1



.

(33)

CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 24 Using Proposition 1 we can extend the action of PSL(2, Z[i]) to H

k,d

by

(H

k,d

|

1−k

γ)(z) = (rz + s)

k−1

(rz + s)

k−1

H

k,d

 pz + q rz + s



(3.7)

= X

∆(h)=d a<0<γt(h)(z,1)



z 1 γ

t

(h)  ¯ z 1



k−1

(3.8)

where γ = p q r s

 .

Next we prove some properties of H

k,d

and show that it is invariant under the action of some special elements of PSL(2, Z[i]).

Lemma 1. 1. H

k,d

(−z) = H

k,d

(z), H

k,d

(¯ z) = H

k,d

(z) and H

k,d

(iz) = H

k,d

(z).

2. H

k,d

(z + 1) = H

k,d

(z) and H

k,d

(z + i) = H

k,d

(z).

Proof. We continue to let the

 a b/2

¯ b/2 c



be the matrix corresponding to the binary Hermitian form h. Then

H

k,d

(−z) = X

∆(h)=d a<0<h(z,1)

h(−z, 1)

k−1

= X

|b|2−4ac=d a<0

max 0,



az ¯ z + 1

2 (−bz − ¯ b¯ z) + c



k−1

!

= X

|b|2−4ac=d a<0

max 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



k−1

!

since if b is a solution to |b|

2

− 4ac = d then so are −b, −¯b. The proofs of H

k,d

(¯ z) =

H

k,d

(z) and H

k,d

(iz) = H

k,d

(z) are similar since b can be replaced by ¯ b or ±ib.

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 25

As for the proof of H

k,d

(z + 1) = H

k,d

(z), notice that H

k,d

(z + 1) = (H

k,d

|

1−k

T )(z) where T = 1 1

0 1



. For ease of notation let us omit writing z 1 and  ¯ z 1

 when we are computing the action of T . So,

(H

k,d

|

1−k

T )(z) = X

∆(h)=d a<0<Tt(h)(z,1)

1 0 1 1

  a b/2

¯ b/2 c

 1 1 0 1



k−1

= X

∆(h)=d a<0<Tt(h)(z,1)

 a a + b/2

a + ¯ b/2 a + ¯ b/2 + b/2 + c



k−1

.

Note that

 a a + b/2

a + ¯ b/2 a + ¯ b/2 + b/2 + c



is just another binary Hermitian matrix with discriminant d and negative first entry. Thus H

k,d

is invariant under the action of T . The fact that H

k,d

(z+i) = H

k,d

(z) can be proved in the same way by computing the action of T

i

= 1 i

0 1



on H

k,d

.

We define the following numbers which are the values of H

k,d

at z = 0:

ω

k,d

: = X

b21+b22−4ac=d a<0<c

c

k

(3.9)

= X

b21+b22<d b1+b2≡d(mod 2)

σ

k

 d − b

21

− b

22

4



. (3.10)

where b = b

1

+ ib

2

∈ Z[i]. In [17] it is shown that, for positive fundamental discrimi-

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 26 nant D and b ranging over integers,

X

|b|< D b≡D(mod 2)

σ

1

 D − b

2

4



= −5L(−1, χ

D

) (3.11)

X

|b|< D b≡D(mod 2)

σ

3

 D − b

2

4



= L(−3, χ

D

). (3.12)

These two numbers are the values of F

k,D

at zero when k = 2, 4. Using (3.11) and (3.12) we can write the values of ω

2,d

and ω

4,d

as a sum of the values of the Dirichlet’s L-functions whenever d − b

21

or d − b

22

is a fundamental discriminant. However, d − b

21

or d − b

22

may not be a fundamental discriminant as b

1

and b

2

range over the solutions of the equation b

21

+ b

22

− 4ac = d with a < 0 < c. Below are the two examples where d − b

21

or d − b

22

stays a fundamental discriminant and where it doesn’t as b

1

and b

2

range over the solutions of the equation b

21

+ b

22

− 4ac = d with a < 0 < c.

Example 1: Let d = 14, then the equation b

21

+b

22

−4ac = d has the following quadru-

plets (a, b

1

, b

2

, c) as solutions: (−1, ±1, ±3, 1), (−1, ±3, ±1, 1), (−1, ±1, ±1, 3), (−3, ±1, ±1, 1).

Since d − b

21

or d − b

22

is a fundamental discriminant in this case we can write ω

2,14

and ω

4,14

as:

ω

2,14

= 2F

2,13

(0) + 2F

2,5

(0)

= −10(L(−1, χ

13

) + L(−1, χ

5

)),

(36)

CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 27 and

ω

4,14

= 2F

4,13

(0) + 2F

4,5

(0)

= 2(L(−3, χ

13

) + L(−3, χ

5

)).

Example 2: Let d = 46, then one of the solutions to the equation b

21

+ b

22

− 4ac = d is the quadruplet (−1, ±1, ±1, 11) and in this case d − b

21

or d − b

22

is 45 which is not a fundamental discriminant. So, when d = 46 we are not able to write ω

2,14

and ω

4,14

just as the sum of L-values.

Similar to F

k,D

, H

k,d

is also constant for k = 2, 4. The poof uses similar ideas as in Zagier’s case. We first show that for k = 2, 4, H

k,d

is constant when z ∈ Q(i) and using the continuity of H

k,d

we conclude that H

2,d

and H

4,d

are constant for all z ∈ C.

Lemma 2. Let z ∈ Q(i) and let S = 0 −1 1 0



∈ PSL(2, Z[i]). Then

(H

2,d

|

−1

S)(z) − H

2,d

(z) = ω

2,d

(z ¯ z − 1). (3.13)

Proof. When z ∈ Q(i), H

2,d

(z) is a finite sum. To see this, suppose h(z, 1) = az ¯ z + 1

2 (bz +¯ b¯ z) + c occurs in the sum for H

2,d

(z) when z = p q + i r

s with p, q, r, s ∈ Z.

Then using the identity d = |¯ b + 2az|

2

− 4ah(z, 1) one gets,

dq

2

s

2

= (s(2ap + b

1

q))

2

+ (q(2ar − b

2

s))

2

+ 4|a|(a(p

2

s

2

+ r

2

q

2

) + qs(b

1

ps − b

2

rq) + cq

2

s

2

)

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CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 28

which bounds a, b

1

, b

2

, c. Thus only finitely many h(z, 1) appear in the sum for H

2,d

(z) when z ∈ Q(i). On the other hand computing (H

2,d

|

−1

S)(z) − H

2,d

(z) directly gives,

= z ¯ zH

2,d

 −1 z



− H

2,d

(z)

= z ¯ zH

2,d

 1 z



− H

2,d

(z)

= z ¯ z X

|b|2−4ac=d a<0

max

 0,  a

z ¯ z + 1 2

 b z +

¯ b

¯ z

 + c



− X

|b|2−4ac=d a<0

max

 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



= X

|b|2−4ac=d a<0

max

 0,

 a + 1

2 (b¯ z + ¯ bz) + cz ¯ z



− X

|b|2−4ac=d a<0

max

 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



= X

|b|2−4ac=d c<0

max

 0,



az ¯ z + 1

2 (b¯ z + ¯ bz) + c



− X

|b|2−4ac=d a<0

max

 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



.

Notice that the terms with a and c both negative in the two previous sums cancel and d not being norm in Z[i] implies that a 6= 0 and c 6= 0 which gives

= X

|b|2−4ac=d c<0<a

max

 0,



az ¯ z + 1

2 (b¯ z + ¯ bz) + c



− X

|b|2−4ac=d a<0<c

max

 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



.

Since max(0, X) = − min(0, −X) for any X ∈ R we get

= X

|b|2−4ac=d c<0<a

max

 0,



az ¯ z + 1

2 (b¯ z + ¯ bz) + c



+ X

|b|2−4ac=d a<0<c

min

 0,



−az ¯ z − 1

2 (bz + ¯ b¯ z) − c



= X

|b|2−4ac=d c<0<a

max

 0,



az ¯ z + 1

2 (b¯ z + ¯ bz) + c



+ X

|b|2−4ac=d c<0<a

min

 0,



az ¯ z + 1

2 (bz + ¯ b¯ z) + c



.

(38)

CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 29 Finally, because max(0, X) + min(0, X) = X for any X ∈ R we get

(H

2,d

|

−1

S)(z) − H

2,d

(z) = X

|b|2−4ac=d c<0<a



az ¯ z + bz 2 +

¯ b¯ z 2 + c



. (3.14)

Since the equation |b|

2

− 4ac = d and the inequality c < 0 < a bound a, |b|, c, there are only finitely many functions, independent of z, contributing to the last sum.

The coefficient of z and ¯ z is zero because if b and ¯ b are solutions to |b|

2

− 4ac = d then so are −b and −¯ b. It is easy to see the coefficient of z ¯ z term is ω

2,d

and the coefficient of the constant term is −ω

2,d

because c < 0. In other words, we have

(H

2,d

|

−1

S)(z) − H

2,d

(z) = ω

2,d

(z ¯ z − 1).

Theorem 8. For every positive integer d ≡ 0, 1, 2 (mod 4) that is not a norm of a Gaussian integer, H

2,d

(z) has a constant value ω

2,d

.

Proof. Let H

2,d0

(z) := H

2,d

(z) − ω

2,d

. Then,

(H

2,d0

|

−1

T )(z) = H

2,d0

(z), (H

2,d0

|

−1

T

i

)(z) = H

2,d0

(z)

and using the definition of H

2,d0

(z) and Lemma 2, we have

(H

2,d0

|

−1

S)(z) = (H

2,d

|

−1

S)(z) − ω

2,d

z ¯ z = H

2,d

(z) + ω

2,d

(z ¯ z − 1) − ω

2,d

z ¯ z = H

2,d0

(z).

Since Q(i) has an Euclidean algorithm, any z ∈ Q(i) can be reduced to zero by

(39)

CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 30 applying S, T, and T

i

in finitely many steps. So, we see that

H

2,d0

(z) = H

2,d0

(0) = 0

for all z ∈ Q(i) which implies that H

2,d

(z) = ω

2,d

for all z ∈ Q(i). Assuming the continuity, we conclude that H

2,d

(z) = ω

2,d

for all z ∈ C.

In Lemma 2, we showed that H

2,d

|

−1

(1 − S) is a polynomial in z and ¯ z. Based on the proof of that lemma and equation (3.14) it is natural to define the following polynomials:

Definition 2. For a positive integer d that is not a norm in Z[i] with d ≡ 0, 1, 2 (mod 4) , we define

P

k,d

(z) : = H

k,d

|

1−k

(S − 1)(z) (3.15)

= X

|b|2−4ac=d c<0<a



az ¯ z + bz 2 +

¯ b¯ z 2 + c



k−1

. (3.16)

Proposition 2. The polynomial P

k,d

satisfies 1. P

k,d

(z) = P

k,d

(−z) = P

k,d

(¯ z) = P

k,d

(iz), 2. P

k,d

|

1−k

(S + 1) = 0 and

3. P

k,d

(z + 1) − P

k,d

(z) = |z|

2k−2

P

k,d

 z + 1 z

 .

Proof. Part 1 is clear from the equation (3.16) once we notice that if b is a solution

to |b|

2

− 4ac = d then so are | − b|, |¯b|, |ib|.

(40)

CHAPTER 3. SUMS OF BINARY HERMITIAN FORMS 31 For part 2 one has

P

k,d

|(S + 1) = H

k,d

|(S − 1)(S + 1) = H

k,d

|(S

2

− 1) = 0

since S

2

= 1.

As for part 3 we have

P

k,d

(z + 1) − P

k,d

(z) = |z + 1|

2k−2

H

k,d



− 1 z + 1



− H

k,d

(z + 1) − |z|

2k−2

H

k,d



− 1 z



+ H

k,d

(z)

= |z + 1|

2k−2

H

k,d



− 1 z + 1 + 1



− H

k,d

(z) − |z|

2k−2

H

k,d

 1 z



+ H

k,d

(z)

= |z + 1|

2k−2

H

k,d

 z z + 1



− |z|

2k−2

H

k,d

 1 z + 1



= |z|

2k−2

P

k,d

 z + 1 z



Lemma 3. We have

(H

4,d

|

−3

S)(z) − H

4,d

(z) = P

4,d

(z) = ω

4,d

(z

3

z ¯

3

− 1). (3.17)

Proof. By Definition 2 we have

(H

4,d

|

−3

S)(z) − H

4,d

(z) = P

4,d

(z) = X

|b|2−4ac=d c<0<a



az ¯ z + bz 2 +

¯ b¯ z 2 + c



3

.

Figure

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